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Unformatted text preview: MATH 131B
2ND PRACTICE MIDTERM
Problem 1. State the book’s deﬁnition of:
(a) A complete metric space
(b)
and
(c) Convergence of a series of real numbers
(d) Normed vector space; Banach space
Problem 2. Let be a metric space with a metric . Let
and be two Cauchy sequences in
. Show that
exists. Note: we do not assume that is complete.
is closed if
Problem 3. Let be the usual Eucledian metric on . We say that a subset
whenever
and
, then
. Show that a subset
is complete with
respect to if and only if it is closed.
Problem 4. Let
be given by
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3 ) &
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¤¢
C¦
9 5
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9 @
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9
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qp bhBg1"eBc2bB$W7R1TSRD
i` Y
X f ) d` Y
X a` Y
X V&
U 3 ) & Show that there is a unique point
with the property that
.
Problem 5. State and prove that Banach contraction principle.
Problem 6. Let
and
be norms on the space
of continuous functions on the
, given by:
interval
G 3 r ) r &
q4¨s8Sxw141sSvD
U 3r ) r & 5
utq41sS&
9 3r)r d ) Y
"h
y D
"Qy #y D
Qy d)Y
"
3 &
pSRD ©§
¨¥
`
T U
'¨r "
#y D
Qy
3 &
p'RD U
"Qy
y D
r f f U
ed f ¡
h¢¦¤¢ £¡
f 1 if and only if Show that the two norms are not equivalent.
Problem 7. Let
and
. Show that
and moreover that if this is the case, then
.
Problem 8. State and prove the comparison test. converges, @
i f U f f © §¥
g0¨¦¤¢ £¡ U
ed ...
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This note was uploaded on 09/19/2011 for the course MATH 131B taught by Professor Hitrik during the Winter '08 term at UCLA.
 Winter '08
 hitrik
 Real Numbers, Vector Space

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